Question
From a medium of index of refraction $${{n_1}},$$ monochromatic light of wavelength $$\lambda $$ is incident normally on a thin film of uniform thickness $$L$$ (where $$L > 0.1\lambda $$ ) and index of refraction $${{n_2}}.$$ The light transmitted by the film travels into a medium with refractive index $${{n_3}}.$$ The value of minimum film thickness when maximum light is transmitted if $$\left( {{n_1} < {n_2} < {n_3}} \right)$$ is
A.
$$\frac{{{n_1}\lambda }}{{2{n_2}}}$$
B.
$$\frac{{{n_1}\lambda }}{{4{n_2}}}$$
C.
$$\frac{\lambda }{{4{n_2}}}$$
D.
$$\frac{\lambda }{{2{n_2}}}$$
Answer :
$$\frac{{{n_1}\lambda }}{{4{n_2}}}$$
Solution :
Equation of path difference form maxima in transmission (or weak reflection)
$$\eqalign{
& \Delta {P_{{\text{opt}}}} = 2{n_2}L = \frac{{{\lambda _{{\text{vacuum}}}}}}{2},\frac{{3{\lambda _{{\text{vacuum}}}}}}{2}\,...... \cr
& \Rightarrow 2\left( {\frac{{{n_2}}}{{{n_1}}}} \right)L = \frac{\lambda }{2},\frac{{3\lambda }}{2},...... \cr
& \Rightarrow L = \frac{\lambda }{{4{n_2}}}\,\,\left( {{\text{notice that }}\lambda = {\text{wavelength in medium is related to }}{\lambda _{{\text{vacuum}}}}\,{\text{as,}}\,{\lambda _{{\text{vacuum}}}} = {n_1}\lambda } \right) \cr} $$